Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1312.1 |
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) } |
2 |
|
bnj1312.2 |
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >. |
3 |
|
bnj1312.3 |
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) } |
4 |
|
bnj1312.4 |
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) |
5 |
|
bnj1312.5 |
|- D = { x e. A | -. E. f ta } |
6 |
|
bnj1312.6 |
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) ) |
7 |
|
bnj1312.7 |
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) ) |
8 |
|
bnj1312.8 |
|- ( ta' <-> [. y / x ]. ta ) |
9 |
|
bnj1312.9 |
|- H = { f | E. y e. _pred ( x , A , R ) ta' } |
10 |
|
bnj1312.10 |
|- P = U. H |
11 |
|
bnj1312.11 |
|- Z = <. x , ( P |` _pred ( x , A , R ) ) >. |
12 |
|
bnj1312.12 |
|- Q = ( P u. { <. x , ( G ` Z ) >. } ) |
13 |
|
bnj1312.13 |
|- W = <. z , ( Q |` _pred ( z , A , R ) ) >. |
14 |
|
bnj1312.14 |
|- E = ( { x } u. _trCl ( x , A , R ) ) |
15 |
6
|
simplbi |
|- ( ps -> R _FrSe A ) |
16 |
5
|
ssrab3 |
|- D C_ A |
17 |
16
|
a1i |
|- ( ps -> D C_ A ) |
18 |
6
|
simprbi |
|- ( ps -> D =/= (/) ) |
19 |
5
|
bnj1230 |
|- ( w e. D -> A. x w e. D ) |
20 |
19
|
bnj1228 |
|- ( ( R _FrSe A /\ D C_ A /\ D =/= (/) ) -> E. x e. D A. y e. D -. y R x ) |
21 |
15 17 18 20
|
syl3anc |
|- ( ps -> E. x e. D A. y e. D -. y R x ) |
22 |
|
nfv |
|- F/ x R _FrSe A |
23 |
19
|
nfcii |
|- F/_ x D |
24 |
|
nfcv |
|- F/_ x (/) |
25 |
23 24
|
nfne |
|- F/ x D =/= (/) |
26 |
22 25
|
nfan |
|- F/ x ( R _FrSe A /\ D =/= (/) ) |
27 |
6 26
|
nfxfr |
|- F/ x ps |
28 |
27
|
nf5ri |
|- ( ps -> A. x ps ) |
29 |
21 7 28
|
bnj1521 |
|- ( ps -> E. x ch ) |
30 |
7
|
simp2bi |
|- ( ch -> x e. D ) |
31 |
5
|
bnj1538 |
|- ( x e. D -> -. E. f ta ) |
32 |
1 2 3 4 5 6 7 8 9 10 11 12
|
bnj1489 |
|- ( ch -> Q e. _V ) |
33 |
7 15
|
bnj835 |
|- ( ch -> R _FrSe A ) |
34 |
1 2 3 4 5 6 7 8 9 10
|
bnj1384 |
|- ( R _FrSe A -> Fun P ) |
35 |
33 34
|
syl |
|- ( ch -> Fun P ) |
36 |
1 2 3 4 5 6 7 8 9 10
|
bnj1415 |
|- ( ch -> dom P = _trCl ( x , A , R ) ) |
37 |
35 36
|
bnj1422 |
|- ( ch -> P Fn _trCl ( x , A , R ) ) |
38 |
1 2 3 4 5 6 7 8 9 10 11 12 36
|
bnj1416 |
|- ( ch -> dom Q = ( { x } u. _trCl ( x , A , R ) ) ) |
39 |
1 2 3 4 5 6 7 8 9 10 11 12 35 38 36
|
bnj1421 |
|- ( ch -> Fun Q ) |
40 |
39 38
|
bnj1422 |
|- ( ch -> Q Fn ( { x } u. _trCl ( x , A , R ) ) ) |
41 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 37 40
|
bnj1423 |
|- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) ) |
42 |
14
|
fneq2i |
|- ( Q Fn E <-> Q Fn ( { x } u. _trCl ( x , A , R ) ) ) |
43 |
40 42
|
sylibr |
|- ( ch -> Q Fn E ) |
44 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
bnj1452 |
|- ( ch -> E e. B ) |
45 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 32 41 43 44
|
bnj1463 |
|- ( ch -> Q e. C ) |
46 |
45 38
|
jca |
|- ( ch -> ( Q e. C /\ dom Q = ( { x } u. _trCl ( x , A , R ) ) ) ) |
47 |
1 2 3 4 5 6 7 8 9 10 11 12 46
|
bnj1491 |
|- ( ( ch /\ Q e. _V ) -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) |
48 |
32 47
|
mpdan |
|- ( ch -> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) |
49 |
48 4
|
bnj1198 |
|- ( ch -> E. f ta ) |
50 |
31 49
|
nsyl3 |
|- ( ch -> -. x e. D ) |
51 |
29 30 50
|
bnj1304 |
|- -. ps |
52 |
6 51
|
bnj1541 |
|- ( R _FrSe A -> D = (/) ) |
53 |
5 52
|
bnj1476 |
|- ( R _FrSe A -> A. x e. A E. f ta ) |
54 |
4
|
exbii |
|- ( E. f ta <-> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) |
55 |
|
df-rex |
|- ( E. f e. C dom f = ( { x } u. _trCl ( x , A , R ) ) <-> E. f ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) ) |
56 |
54 55
|
bitr4i |
|- ( E. f ta <-> E. f e. C dom f = ( { x } u. _trCl ( x , A , R ) ) ) |
57 |
56
|
ralbii |
|- ( A. x e. A E. f ta <-> A. x e. A E. f e. C dom f = ( { x } u. _trCl ( x , A , R ) ) ) |
58 |
53 57
|
sylib |
|- ( R _FrSe A -> A. x e. A E. f e. C dom f = ( { x } u. _trCl ( x , A , R ) ) ) |